3D to 2D projections for cartography

The Mecator projection takes a sphere and projects it onto a cylinder.

I am drawing a world map for a project, but I need to project the sphere differently. My map is a rectangle where the midline of the long axis is a line that crosses each longitude and latitude exactly once (a type of loxodrome or rhumb line).

I need a formula that maps each cartesian coordinate on my map to spherical coordinates on the globe. In this way, I can draw the longitude and latitude lines on my map with precision.

Is this the right subforum that can help? If not please redirect me. Thanks in advance.